Igor - I wouldn't call this a "dictionary order", which is another name for ["lexicographic order"](https://en.wikipedia.org/wiki/Lexicographical_order). In the lexicographical order, the word "at" comes before the word "ba" because a \\(\le\\) b, _even though t\\( \le \\) a is false_: the first letter takes precedence.

As for your main question, we're discussing it [here in the comments to Lecture 22](https://forum.azimuthproject.org/discussion/comment/18302/#Comment_18302) now, so let's continue talking there. If want to ask me about some specific alternative possibilities for the definition of "monoidal preorder", or the difference between the lexicographic order and the product order, that would be great!